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On products of \(k\) atoms. (English) Zbl 1184.20051

Let \(R\) be a Noetherian integral domain and denote by \(V_k(R)\) the set of all integers \(m\) with the property that there exists an element of \(R\) having factorization into irreducibles of length \(k\) and \(m\). The authors show that for a large class of domains the set \(V_k(R)\) is very close to an arithmetic progression. They work actually in a more general setting, dealing with factorizations in atomic monoids, i.e. commutative cancellative semigroups with unit element in which every non-unit is a product of irreducible elements.

MSC:

20M14 Commutative semigroups
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
11R27 Units and factorization
13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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References:

[2] Baginski P, Chapman ST, Hine N, Paixao J (2008) On the asymptotic behavior of unions of sets of lengths in atomic monoids. Involve, to appear
[9] Foroutan A, Geroldinger A (2005) Monotone chains of factorizations in C-monoids. In: Chapman ST (ed) Arithmetical Properties of Commutative Rings and Monoids, Lect Notes Pure Appl Math 241, pp 99–113. Boca Raton, FL: Chapman & Hall/CRC · Zbl 1095.20040
[10] Freeze M, Geroldinger A (2008) Unions of sets of lengths. Funct Approximotio, Comment Math, to appear · Zbl 1228.20046
[22] Schmid WA (2007) A realization theorem for sets of lengths. Manuscript
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